Parking Spaces

TL;DR

This paper introduces two new types of parking spaces for reflection groups, generalizes their definitions, and conjectures their isomorphism, connecting various combinatorial and algebraic structures.

Contribution

It defines noncrossing and algebraic parking spaces for real reflection groups, extending previous concepts and proposing a unifying conjecture about their isomorphism.

Findings

Both new parking spaces are isomorphic to the standard space in the crystallographic case.

Evidence and partial proofs support the main conjecture.

The work links Catalan combinatorics with reflection group theory.

Abstract

Let $W$ be a Weyl group with root lattice $Q$ and Coxeter number $h$. The elements of the finite torus $Q/(h+1)Q$ are called the $W$-{\sf parking functions}, and we call the permutation representation of $W$ on the set of $W$-parking functions the (standard) $W$-{\sf parking space}. Parking spaces have interesting connections to enumerative combinatorics, diagonal harmonics, and rational Cherednik algebras. In this paper we define two new $W$-parking spaces, called the {\sf noncrossing parking space} and the {\sf algebraic parking space}, with the following features: 1) They are defined more generally for real reflection groups. 2) They carry not just $W$-actions, but $W\times C$-actions, where $C$ is the cyclic subgroup of $W$ generated by a Coxeter element. 3) In the crystallographic case, both are isomorphic to the standard $W$-parking space. Our Main Conjecture is that the two new parking spaces are isomorphic to each other as permutation representations of $W\times C$. This conjecture ties together several threads in the Catalan combinatorics of finite reflection groups. We provide evidence for the conjecture, proofs of some special cases, and suggest further directions for the theory.